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CHAPTER XVI. 

 THEEE CLASSICAL GEOMETRICAL PROBLEMS. 



Among the more interesting geometrical problems of anti- 

 quity are three questions which attracted the special attention 

 of the early Greek mathematicians. Our knowledge of geometry 

 is derived from Greek sources, and thus these questions have 

 attained a classical position in the history of the subject. The 

 three questions to which I refer are (i) the duplication of a 

 cube, that is, the determination of the side of a cube whose 

 volume is double that of a given cube ; (ii) the trisection of an 

 angle; and (iii) the squaring of a circle, that is, the deter- 

 mination of a square whose area is equal to that of a given 

 circle — each problem to be solved by a geometrical construction 

 involving the use of straight lines and circles only, that is, by 

 Euclidean geometry. 



This limitation to the use of straight lines and circles implies 

 that the only instruments available in Euclidean geometry are 

 compasses and rulers. But the compasses must be capable of 

 opening as wide as is desired, and the ruler must be of un- 

 limited length. Further the ruler must not be graduated, for 

 if there were two fixed marks on it we can obtain constructions 

 equivalent to those obtained by the use of the conic sections. 



With the Euclidean restriction all three problems are in- 

 soluble*. To duplicate a cube the length of whose side is a, 



* See F. C. Klein, Vortrage fiber ausgewahlte Fragen der Elementargeometrie, 

 Leipzig, 1895; and F. O. Texeira, Sur les Problemes cSl&bres de la Ge'ome'trie 

 Mimentaire non resoluble) avec la Regie et le Compos, Ooimbra, 1915. It is said 

 that the earliest rigorous proof that the problems were insoluble by Euclidean 

 geometry was given by P. L. Wantzell in 1837. 



D. R. 22 



